linear_algebra/polynom_for_points: ZeroDivisionError on valid points containing x=0 (Gaussian elimination without pivoting)

[aimasteracc]

Description

linear_algebra/src/polynom_for_points.py::points_to_polynomial crashes with ZeroDivisionError for valid inputs that include a point with x = 0. It solves the interpolation system with Gaussian elimination without pivoting, so a zero on the diagonal divides by zero:

for count in range(x):
    for number in range(x):
        if count == number:
            continue
        fraction = matrix[number][count] / matrix[count][count]   # /0 when matrix[count][count] == 0
        ...

matrix[count][count] = coordinates[count][0] ** (x - (count + 1)). When an x-coordinate is 0, that diagonal entry is 0 ** positive = 0, so the very first elimination step divides by zero. The input passes every validation in the function (non-empty, all pairs length-2, distinct points, distinct x-values) — it is a genuinely fittable set — and then crashes in the solver.

Steps to reproduce (verified)

from linear_algebra.src.polynom_for_points import points_to_polynomial

points_to_polynomial([[0, 1], [1, 2], [2, 5]])   # f(x) = x^2 + 1 passes through these
# -> ZeroDivisionError: division by zero

points_to_polynomial([[0, 0], [1, 1], [2, 4]])    # f(x) = x^2
# -> ZeroDivisionError: division by zero

The existing doctests only use x-values 1, 2, 3, so the x = 0 case is uncovered.

Expected behavior

For [[0, 1], [1, 2], [2, 5]] it should return the fitting polynomial (f(x)=x^2*1.0+x^1*0.0+x^0*1.0), not crash. Any set of points with distinct x-values has a unique interpolating polynomial.

Actual behavior

ZeroDivisionError whenever any point has x = 0 (and generally whenever a zero pivot appears on the diagonal during elimination).

Suggested fix

Use partial pivoting: before each elimination step, swap in a row whose matrix[r][count] is non-zero (largest magnitude). This removes the zero-pivot crash and also improves numerical stability. A doctest with an x = 0 point (e.g. points_to_polynomial([[0, 1], [1, 2], [2, 5]])) should be added to cover it.